Working Paper/Document de travail 2015-11

Fourier Inversion Formulas for Multiple-Asset Option Pricing

by Bruno Feunou and Ernest Tafolong

2

Bank of Canada Working Paper 2015-11

March 2015

Fourier Inversion Formulas for Multiple-Asset Option Pricing

by

Bruno Feunou1 and Ernest Tafolong2

1Financial Markets Department Bank of Canada

Ottawa, Ontario, Canada K1A 0G9 Corresponding author: feun@bankofcanada.ca

2National Bank of Canada

1155, rue Metcalfe 5e étage Montréal, Quebec H3B 4S9

ernest.tafolong@bnc.ca

Bank of Canada working papers are theoretical or empirical works-in-progress on subjects in economics and finance. The views expressed in this paper are those of the authors.

No responsibility for them should be attributed to the Bank of Canada or the National Bank of Canada.

ISSN 1701-9397 © 2015 Bank of Canada

ii

Acknowledgements

We thank Guillaume Nolin, Maren Hansen, Glen Keenleyside and Foutse Khomh for comments and suggestions. A previous version of this paper circulated under the title “Pricing Multiple Triggers Contingent Claims.” Remaining errors are ours.

iii

Abstract

Plain vanilla options have a single underlying asset and a single condition on the payoff at the expiration date. For this class of options, a well-known result of Duffie, Pan and Singleton (2000) shows how to invert the characteristic function to obtain a closed-form formula for their prices. However, multiple-asset and multiple-condition derivatives such as rainbow options cannot be priced within this framework. Utilizing inversion of the Fourier transform – and resorting to neither the Black-Scholes framework nor the affine models settings – the authors provide an analytical solution for options whose payoffs depend on two or more conditions. Numerical experiments based on the multiple-asset and multiple-condition derivatives are provided to illustrate the usefulness of the proposed approach.

JEL classification: G12 Bank classification: Asset pricing

Résumé

Les options classiques n’ont qu’un sous-jacent, et leur valeur à l’échéance est déterminée par une seule condition. Un des apports bien connus de l’étude de Duffie, Pan et Singleton (2000) est d’avoir montré comment utiliser la transformée inverse de la fonction caractéristique pour obtenir une formule analytique des prix dans cette classe d’options. Or, les prix des dérivés à plusieurs sous-jacents et conditions, tels que les options arc-en-ciel, ne peuvent être évalués à l’intérieur de ce cadre. En s’appuyant sur la formule d’inversion de la transformée de Fourier, mais sans recourir au modèle de Black et Scholes ni aux modèles affines, les auteurs proposent une solution analytique pour calculer le prix d’options dont la valeur à l’échéance est déterminée par deux conditions ou plus. Ils procèdent à des expériences numériques afin d’illustrer l’utilité de l’approche proposée dans le cas des dérivés à plusieurs sous-jacents et conditions.

Classification JEL : G12 Classification de la Banque : Évaluation des actifs

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Non-Technical Summary Derivatives (or options) pricing is an important topic for both academics and practitioners. Two approaches are generally considered when pricing a derivative, namely numerical methods and closed-form expressions. There are numerous advantages of having closed-form derivatives prices, among them: (1) they enable a quick and efficient computation of the price, (2) they can be used to evaluate derivatives’ price sensitivity to a given parameter, and (3) they are useful in qualitative analysis. Unfortunately, combining closed-form and realistic dynamics (on assets underlying the derivative) has proven very challenging.

Other works in the literature evaluate derivatives written on a single stock. However, this paper proposes a closed-form approach to evaluate derivatives written on several stock prices. Besides providing users with an analytical solution to multivariate derivative pricing, we decompose the resulting prices and interpret each component.

In a setting where a derivative written on two assets has been evaluated, we can show that the resulting price has three intuitive components. The first, termed the “constant’’ component, changes mostly with the price of the underlying assets. The “constant” component can be assimilated to the premium that derivatives buyers are willing to pay to hedge against variations in the level of the stock. The second is referred to as the “volatility’’ component, and is found to be mostly correlated to the average volatility across underlying assets. It can be interpreted as the premium paid by options buyers to hedge against variation in individual volatilities. The final component is designated as the “correlation’’ component, since it co-moves with correlation among underlying assets. Hence we interpret this last piece as the premium paid by options buyers to hedge against variations in the correlation of underlying assets.

The paper highlights several interesting methodological aspects, but it leaves out some real-world issues such as risk-neutral parameters calibration and risk-neutral distribution fitting. Our approach would be useful in these and other areas as more options data become available.

1 IntroductionIn a contingent claims valuation, the lack of closed-form solutions for derivativesmay undermine its practical use and pose significant implementation hurdles. Theanalytical formula embedded in the Black-Scholes settings1 substantially lessensthe computational burden of options pricing. Unfortunately, the closed-form pricederived under the Black-Scholes assumptions cannot be carried over, even for one-dimensional derivatives, when accounting for well-known stylized facts about re-turns.2 It is much more challenging to price options with several dimensions andmultiple payoff conditions, since the high dimensionality often raises the computa-tional cost.

Furthermore, adding flexibility to benchmark models such as Black-Scholesoften comes with a cost: the loss of its applicability. Heston (1993) shows how toget a closed-form price to European options when the underlying asset features bothstochastic volatility and the leverage effect. Duffie et al. (2000) generalize Heston’sframework to any univariate time-series model with a closed-form characteristicfunction. Yet, this paper goes beyond the univariate setting, by proposing an ana-lytical solution to option pricing within a multivariate framework where, in additionto time variations of individual volatilities, we also have time-varying correlations.When departing from the Black-Scholes valuation framework, professionals oftenresort to numerical schemes and simulations to value exotic options (e.g., Glasser-man, 2003; Duffy, 2009). However, numerical techniques and simulations sharecommon flaws: dimensionality3 and the high computational burden.

This research underscores the usefulness of the Fourier-Stieltjes transform4

in deriving a semi-analytical solution where the computational task of options pric-ing comes down to the evaluation of the Laplace transform. The relevance of oursemi-analytical solution is based on two main arguments. First, its empirical ap-plicability is enhanced by virtue of an analytical formula, which can be used incalibration problems. Thus, some parameters of underlying asset distribution (im-plied parameters) can be recovered directly from available options prices. More-over, contrary to the historical data on underlying assets, options data are forwardlooking, since they can be used to infer the underlying asset distribution. Thus we

1The main assumptions of the Black-Scholes model are: a continuous stochastic process, con-stant volatility, the efficient market hypothesis and log-normal distributions.

2Empirical facts that do not support Black-Scholes assumptions include: stochastic volatility,jumps, fat tails, skewness and leverage effects.

3It is worth noting that the Monte Carlo simulations do not exhibit the curse of dimensionality.However, as we will show in our empirical section, the high computation time in the Monte Carlosimulations is a concern.

4In the remainder of the paper, we will be using Laplace transform, characteristic function andFourier-Stieltjes transform interchangeably.

1

can use our closed-form options prices to fit observed prices, and provide new in-sights which are missing from historical underlying assets data.5 Second, one ofthe greatest virtues of the Laplace transform is its versatility. Indeed, unlike nu-merical methods, the Fourier transform can be implemented in a numerically stableway (Davies, 1973). In fact, whenever the characteristic function of the state vari-able exists in closed form, how to infer a closed-form formula for options pricesis well known (Duffie et al., 2000; Dufresne et al., 2009). However, to the bestof our knowledge, inferring multiple-condition contingent claims with the Fouriertransform has not yet been addressed in the literature. This omission is particularlyglaring, since: (a) basket options are by far the most liquid contingent claims for in-vestors,6 and (b) the correlation and volatility parameters are the crucial drivers forpricing basket options (see Buraschi et al., 2014; Qu, 2010), particularly under ad-verse market conditions. In order to fill this void, this paper provides the frameworkto value options on multiple-asset and multiple-payoff conditions.

Our work is closely related to that of Duffie et al. (2000), where the closed-form price for a single condition is derived. Those authors exploit the well-establishedexponential affine form of the characteristic function embedded in the affine pro-cess in order to compute option prices. Although the affine processes have attractedmuch attention from both professional and academic audiences, for the theory de-veloped in this paper, the only requirement is the existence of the Laplace transformin closed form.7 For the empirical investigation, we use structured products writtenon the NYSE and NASDAQ indices. Since both indices are market-value weighted,they can be used in risk management and portfolio optimizations.

To illustrate concretely how our pricing formulas work, we develop a sim-ple, theoretically grounded and broadly applicable multivariate model (affine real-ized variance, or ARV) that captures individual and joint dynamics in several stock

5For example, Navatte and Villa (2000) demonstrate that the recovered implied moments of theCAC 40 significantly improved out-of-sample pricing performance; Buss and Vilkov (2012) find thatusing options data to construct implied factor betas significantly outperforms historical betas in thecapital asset pricing model; Kempf et al. (2014) report substantial portfolio gains when relying solelyon the plain vanilla options to calibrate a covariance matrix; Buraschi et al. (2014) demonstrate theusefulness of implied correlations in the factor returns construction; and van Binsbergen et al. (2012)use well-known put-call parity (Stoll, 1969) to construct a term structure analog of dividend yieldsfor the market portfolio by calibrating implied dividend yields from the option contracts.

6In Canada, for example, Structured-Retails-Products (2013) report that the notional amount ofoptions written on basket was 2.865 billion CAD, whereas that of single shares was only 16 millionCAD in 2013.

7Many papers compute the closed-form expression of the characteristic function: notably, theaffine models literature, which includes Bates (1996), Bakshi et al. (1997), Duffie et al. (2003),and Christoffersen et al. (2006). Other papers, such as Feunou and Meddahi (2009), provide ageneral framework that characterizes the infinite-order affine model. Some prominent examples ofnon-affine models are Ahn and Gao (1999); Feunou and Fontaine (2010).

2

market returns. Because the ARV model is affine, its conditional characteristicfunction at any horizon is available in closed form. We find conclusive evidencethat options prices in the ARV model are similar to prices in the standard bench-mark, non-affine dynamic conditional correlation (DCC) model of Engle (2000).Within the ARV model, Monte Carlo simulations are used as an alternative to ourclosed-form approach. While both approaches provide similar options prices, thecomputation time for the Monte Carlo approach is about twelve times that of ours.Furthermore, beyond elegant mathematical expression, the advantage of transformanalysis is its ability to break down multivariate options prices into intuitive com-ponents. Thus, this paper represents a significant step toward better understandingmultivariate options pricing drivers and their impacts on effective risk assessment.

The remainder of this paper is organized as follows. Section 2 is devotedto the set-up. Section 3 states the main results and section 4 introduces the ARVmodel. Section 5 describes how to use our theoretical framework to price rainbowoptions within the ARV model. Section 6 provides concluding remarks. To preservethe main thread of the paper, we have placed many of the technical proofs anddetails in appendices.

2 The set-up

2.1 Notation and setting

We consider a financial market with derivatives depending on multivariate statevariables:

• Xt = (X1,X2, . . . ,Xn)t is a column vector of n state variables at time t.• (Xi)t≥0 denotes the process of asset i, i = 1,2, . . . ,n defined in D⊆ Rn.• u = (u1,u2, . . . ,un) ∈ Cn is a row vector of complex variables with the same

dimension as the state vector Xt at each time t.• T is the option’s maturity.

Given available information up to time t, Ft , for t ≤ T , we assume that the condi-tional discounted moment-generating function of XT can be expressed as

ψ(u,Xt , t,T ) = Et

[exp(−∫ T

tR(Xs)ds

)eu.XT

], (1)

where Et denotes the conditional expectation given the information set (Ft) up to t.In other words, the conditional expectation is the function of u,Xt , t,T . We wouldlike to emphasize that the class of affine jump diffusions from Duffie et al. (2000)

3

is a special case of our framework where ψ(·) can be computed using an affinemodels framework.8

2.2 Common multiple-condition contingent claims

Below are some examples of rainbow options that are of sufficient interest in thecontext of basket options.9 The pricing formulas for two assets are provided insection 4.3.

Ê “Best/worst of assets or cash,” paying the maximum/minimum of two or sev-eral securities and cash at maturity: max/min(X1,X2, . . . ,Xn,K) (for details,see Johnson, 1987; Martzoukous, 2001).

Ë “Call on max/min,” which entitles the owner to buy the maximum/mini-mum asset at a given strike at expiry: max(max/min(X1,X2, . . . ,Xn)−K)(see Johnson, 1987).

Ì “Put on max/min,” giving the holder the right to sell the maximum/min-imum asset at a given strike at expiry: max(K−max/min(X1,X2, . . . ,Xn))(see Johnson, 1987).

Í “Exchange one asset for another and earn the spread between the two,” whichenables the long-position investor to sell/buy an asset strike price given bythe price of another: max/min (X1−X2,0) (see Margrabe, 1978; Gay andManaster, 1984).

3 Main resultsIn this section, we review analytical solutions of increasing difficulty. Firstly, insection 3.1 we address the benchmark single condition on the terminal payoff. Sec-ondly, within the multiple-condition framework, we introduce in section 3.2 thebenchmark bivariate case. Finally, the extension to more than two conditions isprovided in section 3.3.

3.1 One-condition derivatives pricing

In this section, we review one condition in the option payoff, as in Duffie et al.(2000). Given (x,T,a,b) ∈ D× [0,+∞]×Rn×Rn , let Ga,b (.,x,T ) : R→ R+ be

8We implicitly assume that the expectation is well defined. This assumption is fully covered onpage 1351 of Duffie et al. (2000).

9Although this section focuses on popular rainbow options, it is important to grasp that otherexamples can be amended to cater for multiple conditions on the payoff.

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defined as follows:

Ga,b(y,Xt ,T ) = Et

[exp(−∫ T

tR(Xs)ds

)ea.XT 1bXT≤y

]. (2)

We recall the following Proposition 2.2 from Duffie et al. (2000), whichinvolves a single condition on the payoff.10

Proposition 1

Ga,b(y,x,T )=ψ(a,x, t,T )

2+

14π

∫ +∞

−∞

eivyψ(a− ivb,x, t,T )− e−ivyψ(a+ ivb,x, t,T )iv

dv.

(3)

Proof. See Appendix A of Duffie et al. (2000).Obviously, the single-condition payoff examined previously cannot be used

to price multiple-condition options. Before we rush to tackle the multiple-conditionpayoff, let us get more comprehensive insights from two conditions, because theyemerge as a special case of the multiple-condition payoff.

3.2 Semi-closed-form for two conditions

In this section, we provide a formula to price derivatives with two conditions on theoptions payoff. For expositional purposes, we provide details for the two conditionsin Appendix A.

Given (x,T,a,b1,b2)∈D×[0,+∞]×Rn×Rn×Rn , let us define Ga,b1,b2 (.,x,T ) :R2→ R+ as follows:

Ga,b1,b2(y1,y2,x,T ) = Et

[exp(−∫ T

tR(Xs)ds

)ea.XT 1b1XT≤y11b2XT≤y2

]. (4)

The goal is to compute Ga,b1,b2(y1,y2,x,T ) given by expression (4). We recover asingle-dimension case of Duffie et al. (2000) (Proposition 1) by letting one of y1 ory2 become ∞.

At this stage, we recognize that Monte Carlo simulations can be used fromthe outset of the option payoff. Notwithstanding the issue of complex optionswith double conditions, Monte Carlo simulations and other numerical schemes are

10We can name this contingent claim payoff (2) the "One condition option," since it involves onlyone indicator function.

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more computationally intensive than our Fourier-Stieltjes transform. The Fourier-Stieltjes transform is well defined by

Ga,b1,b2 (v1,v2,Xt ,T ) =∫R2

eiv1y1+iv2y2Ga,b1,b2(dy1,dy2,Xt ,T )

= Et

[exp(−∫ T

tR(Xs)ds

)exp(a+iv1b1+iv2b2)XT

]= ψ(a+ iv1b1 + iv2b2,Xt , t,T ).

(5)

For y1,y2 ∈ R, let us define the expression I{1,2} by the following relation:

I{1,2} ≡∫R2

eiv1y1+iv2y2ψ(a− iv1b1− iv2b2)− e−iv1y1+iv2y2ψ(a+ iv1b1− iv2b2)

(iv1)(iv2)dv1dv2

+

∫R2

−eiv1y1−iv2y2ψ(a− iv1b1 + iv2b2)+ e−iv1y1−iv2y2ψ(a+ iv1b1 + iv2b2)

(iv1)(iv2)dv1dv2.

(6)

The option price formula we have been looking at can then be reduced tothe following proposition.

Proposition 2

Ga,b1,b2(y1,y2,x,T ) =ψ(a,x, t,T )

4+

116π2 I{1,2}

+1

8π

∫ +∞

−∞

eivy2ψ(a− ivb2,x, t,T )− e−ivy2ψ(a+ ivb2,x, t,T )iv

dv

+1

8π

∫ +∞

−∞

eivy1ψ(a− ivb1,x, t,T )− e−ivy1ψ(a+ ivb1,x, t,T )iv

dv.

(7)

Let us denote Re(·) and Im(·), respectively, as the real and the imaginarypart of a complex number. Using sin and cos properties along with the parity of theintegrand, we can derive the following, more compact, formulas for Proposition 2:

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Ga,b1,b2(y1,y2,x,T )

=ψ(a,x, t,T )

4

− 12π

∫ +∞

0

Im[e−ivy1ψ (a+ ivb1)+ e−ivy2ψ (a+ ivb2)

]v

dv

− 12π2

∫ +∞

0

∫ +∞

0

Re[e−iv1y1−iv2y2ψ (a+ iv1b1 + iv2b2)

]v1v2

dv1dv2

− 12π2

∫ +∞

0

∫ +∞

0

Re[e−iv1y1+iv2y2ψ (a+ iv1b1− iv2b2)

]v1v2

dv1dv2.

(8)

Proof. The proof of Proposition 2 relies on two-dimensional differentialcalculus and trigonometric equations (see Appendix A).

In this section, we concentrate on elucidating the case of the two-conditionspayoff, which greatly strengthens our intuition of multiple conditions and proves tobe a useful connection with some derivatives (e.g., secured debt valuation in Changet al., 2006, and cost-of-living contracts in Stulz, 1982). Let us abstract from thetwo-conditions framework above to examine the valuation of multiple-conditioncontingent claims in a more general setting.

3.3 Pricing formulas with more than two conditions

In this section, we investigate the valuation of multiple-condition contingent claimsin the general setting. To characterize a solution effectively, we give the problem alittle more structure:

• Denote E = {1,2, . . . ,m} as a set of m members, where m denotes the exercisedomain; i.e., the number of condition indicator functions determining thepayoff. In the sequel, |.| stands for the cardinality of the members of anysubset of E, and Ek denotes a subset of E with k members (|Ek|= k).• Define E as the power set of E. We have |E|= 2m. For each A ∈ E, Ac is the

complementary of A; in other words, it is the subset of members of E that arenot in A.

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Our goal is to compute the multiple-condition valuation for the following expecta-tion on the terminal payoff:

Ga,b(y,Xt ,T ) = Et

exp(−∫ T

tR(Xs)ds

)ea.XT

m∏j=1

1b jXT≤y j

, (9)

where b j ∈Rn,∀ j = 1,2, . . . ,m. Recall that n is the number of state variables in ourpricing formula, while m is the number of conditions.

Set IA,A ∈ E as follows:

IA =

∫R|A|

∑B⊆A(−1)|B|eiv(B)y(B)−iv(A\B)y(A\B)ψ

(a− iv(B)b(B)+ iv(A\B)b(A\B)

)∏j∈A(iv j)

∏j∈A

dv j,

(10)where A\B is a subset of members in A that are not in B and |B| is the cardinalityof B and v(B)y(B) =

∑j∈B v jy j. The summation

∑B⊆A in (10) is the sum over the

subset of A with |B| members.11

Lemma 1 Let QA be defined by the following expression:

QA ≡∑

B⊆A(−2)|B|eiv(B)(y(B)−z(B))−iv(A\B)(y(A\B)−z(A\B))∏j∈A(iv j)

.

We then have the following relation:

QA = (−2)|A||A|∏j=1

sin(v j(y j− z j

))v j

. (11)

The proof of Lemma 1 rests on the combinatory identity (refer to Appendix B formore details). With the help of Lemma 1, we can state the following lemma.

Lemma 2 Denote as Ga,b(y(A)) the option value where only the yi with i ∈ A arenot +∞; then we have

IA = (−2π)|A|∑B⊆A

(−2)|B|Ga,b(y(B)). (12)

11It is worth noting that for a null set, Ga,b(y(∅)) = Ga,b(

m times︷ ︸︸ ︷+∞,+∞, . . . ,+∞) = ψ(a,x, t,T ). Ac-

cordingly, we set IA = ψ(a,x, t,T ) whenever A is a null set.

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Proof. The proof of Lemma 2 resorts to Lemma 1 (see Appendix C).We are now in a position to price any given number of multiple conditions

on options payoff. We have the following proposition.

Proposition 3 For y = (y1,y2, . . . ,ym) ∈ Rm,

2mGa,b(y1,y2, . . . ,ym,x,T ) =∑A⊆E

1(2π)|A|

IA, (13)

where IA,A ∈ E is given by (10)

The proof of Proposition 3 is given in Appendix D.Proposition 3 is the main theoretical contribution of our paper. It derives the

price of derivatives up to numerical integrations where the only requirement is theexistence of the Fourier transform in closed form.

4 A multivariate affine model for stock pricesIn this section, we introduce the return and realized covariance data, as well asa multivariate model that describes the joint dynamic of several asset returns. Themodel is the multivariate analog of the realized volatility (ARV) introduced in Christof-fersen et al. (2014). The model is estimated in section 5 and used to illustrate howto concretely apply our theory to price rainbow options.12 To avoid excessive de-partures from the main purpose of the current section, we provide only the mainingredients of the pricing and save the technical part of the model for Appendix E.

4.1 The affine realized variance (ARV) model

The affine model has gained widespread acceptance due to its analytical tractability,as well as its flexibility in coping with some stylized facts about returns (time-varying returns, stochastic volatility, volatility risk premium, etc). Thus, in orderto skirt some restrictions implicit in the Black-Scholes framework, the affine modelcan provide a theoretical benchmark for the joint dynamic of returns and volatility.

Rt is a vector of log-returns (Rt = ln(St/St−1)) of dimension n, and RVt a re-alized variance-covariance n×n matrix, both observed at the end of day t. It is wellknown (see Andersen and Andreasen (2000), Andersen et al. (2001) and Forsberg

12But it is worth noting that the framework introduced in this paper is not restricted to the partic-ular choice of data, dynamics or options to be priced.

9

and Bollerslev (2002)) that the distribution of returns standardized by the realizedvariance is Gaussian. Hence we can write

Rt = µt−1 +(RVt−Σt−1)δ +RV 1/2t z1,t ,

where the z1,t are iid and normally distributed (N(0, In)), Σt−1 is the conditionalexpectation of the realized variance

Σt−1 = Et−1 [RVt ] ,

and µt−1 is the conditional expectation of log returns

µt−1 = Et−1 [Rt ] .

We will assume that the shock in the realized variance follows a standardWishart distribution (widely used to model the variance-covariance matrix; see Gourier-oux, 2006; Gourieroux and Sufana, 2010; Buraschi et al., 2010):

RVt = Σt−1 +σ [Wt−Et−1 (Wt)]σ′,

where σ is an n×n matrix, and the Wt are iid and Wishart distributed witha degree of freedom p > n−1 and variance V . Denote Wt ∼Wn(V, p), where V is asymmetric positive definite matrix.

4.2 Conditional expectations

4.2.1 Conditional expectations of the realized variance-covariance matrix

The goal here is to specify the dynamic of Σt−1 = Et−1 [RVt ] , though it is importantto mention that Σt−1 is not only the expectation of the realized variance-covariancematrix, but also the conditional variance-covariance matrix of the returns

Σt−1 = Et−1 [RVt ] =Vart−1 [Rt ] .

Consistent with recent literature (e.g., Shephard and Sheppard, 2010; Christof-fersen et al., 2014), we assume that Σt is updated through RVt ; hence,

Σt = ω +βΣt−1β′+αRVtα

′, (14)

with ω a symmetric, semi-definite positive matrix. This specification is very similarto the GARCH specification, the only difference being that we use the realizedvariance to update the volatility instead of the noisy daily squared-returns. We canfurther express (14) more explicitly (see Appendix E.2).

10

4.2.2 Conditional expectations of log returns

Denote the risk-free rate as r f , the price of risk as λ , and an n−column vector thathas 1 at row i and 0 elsewhere as ei. We have the following proposition.

Proposition 4 If the conditional mean at t−1, µt−1, is chosen such that

Et−1[exp(e′iRt

)] = exp

(r f +λie′iΣt−1ei

),

then

e′iµt−1 = r f +

(λi−

12

)e′iΣt−1ei

+pe′iσV σ′δ +

p2

e′iσV σ′ei

+p2

ln[

det(

In−2(

σ′δe′iσ +

12

σ′eie′iσ

)V)]

.

Proof. The proof of Proposition 4 is given in Appendix E.3.

4.2.3 Moment-generating function

We express the one-step-ahead, conditional-moment-generating function by the fol-lowing proposition.

Proposition 5 The one-step-ahead, conditional-moment-generating function is givenby

Et−1[exp(u′Rt +Tr (θΣt)

)]= exp(A(u,θ)+Tr (B(u,θ)Σt−1)) ,

with

A(u,θ) =

∑ni=1 ui

(r f + pe′iσV σ ′δ + p

2 e′iσV σ ′ei+ p

2 ln[det(In−2

(σ ′δe′iσ + 1

2σ ′eie′iσ)

V)] )

− p2 ln[det(In−2

(12σ ′uu′σ +σ ′δu′σ +σ ′α ′θασ

)V)]

−pu′σV σ ′δ − p2 u′σV σ ′u+Tr (θ (ω− pασV σ ′α ′))

B(u,θ) = β′θβ +α

′θα +

12

uu′+n∑

i=1

eiui

(λi−

12

)e′i.

11

Proof. The proof of Proposition 5 is given in Appendix E.4.1.We also provide the multi-steps-ahead moment-generating function in the

following proposition.

Proposition 6

Et

[exp

(u′

τ∑i=1

Rt+i

)]= exp(C (u;τ)+Tr (D(u;τ)Σt)) ,

with

C (u;1) = A(u,0)D(u;1) = B(u,0) ,

and

C (u;τ +1) = C (u;τ)+A(u,D(u;τ))

D(u;τ +1) = B(u,D(u;τ)) .

Proof. The proof of Proposition 6 is given in Appendix E.4.2. We nextuse the ARV model to illustrate concretely how to evaluate rainbow contracts in thebivariate case (n = 2).

4.3 Bivariate option pricing (n = 2)

In this section, we give the pricing formula for the bivariate case and show how itcan be used to price the examples listed in section 2.2.

4.3.1 Options formulas for the bivariate case

Given that the price of an option is the discounted expectation of future payoff underthe risk-neutral measure Q, option pricing thus does require a change of probabil-ity. That change is routinely done in the literature through the Esscher transform(see Gerber and Shiu, 1994 for more details). We follow a similar pattern, and find

12

that the risk-neutral dynamic is almost the same as the historical up to one parame-ter, λ . Under Q, we should make sure to fix λ = 0, which implies that

EQt−1[exp(Rt)] = exp

(r f).

For a given time t (0 ≤ t ≤ T ), S1t and S2,t are the price at t of two stocks.Define

Xt = (ln(S1t) , ln(S2t))′ . (15)

Our expression (1) implies that

ψ (u;Xt ,Σt) = EQt[exp(−(T − t)r f +u′XT

)]= EQ

t[exp(−(T − t)r f +u′Xt +u′ (XT −Xt)

)]= exp

(−(T − t)r f +u′Xt

)EQ

t[exp(u′ (XT −Xt)

)]= exp

(−(T − t)r f +u′Xt +C (u;T − t)+Tr (D(u;T − t)Σt)

),

where C(u;T − t) and D(u;T − t) are given by Proposition 6, and Σt is given by(14).

Following Proposition 2, we have the following expression:

Ga,b1,b2 (y1,y2;Xt ,Σt)

=ψ (a;Xt ,Σt)

4(16)

− 12π

∫ +∞

0

Im[e−ivy1ψ (a+ ivb1)+ e−ivy2ψ (a+ ivb2)

]v

dv (17)

− 12π2

∫ +∞

0

∫ +∞

0

Re[e−iv1y1−iv2y2ψ (a+ iv1b1 + iv2b2)

]v1v2

dv1dv2

− 12π2

∫ +∞

0

∫ +∞

0

Re[e−iv1y1+iv2y2ψ (a+ iv1b1− iv2b2)

]v1v2

dv1dv2. (18)

As shown in the above formula, the closed-form price can be decomposedinto three components. The first component (16) is the conditional-moment-generatingfunction. We designate it the "Constant-Part" and we expect very little time-seriesvariation. The second component (17) is an integral, which we refer to as the"Volatility-Part." Intuitively, we expect its time-series variation to be mostly drivenby variation in the individual variances. The third and final component (18) is adouble integral designated "Correlation-Part." We conjecture its time-series varia-tion to be mostly driven by variations in the correlation. In the following section,we use the formula to price well-known options.

13

4.3.2 Pricing common bivariate contingent claims

This section provides the pricing formulas in the case of two assets for the optionslisted in section 2.2. Throughout this section, e1 is a vector (1,0) and e2 is a vec-tor (0,1). To ease the notations, we drop the arguments (Xt ,Σt) in the functionGa,b1,b2 (y1,y2;Xt ,Σt).

Ê “Best of assets or cash,” paying the maximum of two securities and cash atmaturity. The payoff is given by

max(S1T ,S2T ,K)

= S1T 1[S1T>S2T ]1[S1T>K]+S2T 1[S2T>S1T ]1[S2T>K]+K1[S1T<K]1[S2T<K]

= ee′1XT 1[lnS2T−lnS1T<0]1[− lnS1T<− lnK]+ ee′2XT 1[lnS1T−lnS2T<0]1[− lnS2T<− lnK]

+ K1[lnS1T<lnK]1[lnS2T<lnK].

Hence, the price of this option at time t is the sum of three quantities:

Et[exp(−r f (T − t)

)max(S1T ,S2T , ,K)

]= Ge1,e2−e1,−e1 (0,− lnK)+Ge2,e1−e2,−e2 (0,− lnK)+KG0,e1,e2 (lnK, lnK) .

Ë “Call on max,” which entitles the owner to buy the maximum of two assets ata given strike at expiry. The payoff here is

max(max(S1T ,S2T )−K,0)= (max(S1T ,S2T )−K)1[K<max(S1T ,S2T )]

= max(S1T ,S2T )1[K<max(S1T ,S2T )]−K1[K<max(S1T ,S2T )]

= (S1T −K)1[K<S1T ]1[S1T>S2T ]+(S2T −K)1[K<S2T ]1[S1T<S2T ].

Then, the price of this option at time t is the sum of four quantities:

Et[exp(−r f (T − t)

)max(max(S1T ,S2T )−K,0)

]= Ge1,e2−e1,−e1 (0,− lnK)−KG0,e2−e1,−e1 (0,− lnK)

+Ge2,e1−e2,−e2 (0,− lnK)−KG0,e1−e2,−e2 (0,− lnK) .

Ì “Call on min,” which entitles the owner to buy the minimum of two assets ata given strike at expiry. The payoff here is

max(min(S1T ,S2T )−K,0)= (min(S1T ,S2T )−K)1[K<min(S1T ,S2T )]

= min(S1T ,S2T )1[K<min(S1T ,S2T )]−K1[K<min(S1T ,S2T )]

= (S2T −K)1[K<S2T ]1[S1T>S2T ]+(S1T −K)1[K<S1T ]1[S1T<S2T ].

14

The price of this option at time t is the sum of four quantities:

Et[exp(−r f (T − t)

)max(max(S1T ,S2T )−K,0)

]= Ge1,e1−e2,−e1 (0,− lnK)−KG0,e1−e2,−e1 (0,− lnK)

+Ge2,e2−e1,−e2 (0,− lnK)−KG0,e2−e1,−e2 (0,− lnK) .

Í “Put on max,” giving the holder the right to sell the maximum of two assetsat a given strike at expiry. The payoff here is

max(K−max(S1T ,S2T ),0)= (K−max(S1T ,S2T ))1[K>max(S1T ,S2T )]

= (K−S1T )1[K>S1T ]1[S1T>S2T ]+(K−S2T )1[K>S2T ]1[S1T<S2T ].

The price of this option at time t is the sum of four quantities:

Et[exp(−r f (T − t)

)max(max(S1T ,S2T )−K,0)

]= −Ge1,e2−e1,e1 (0, lnK)+KG0,e2−e1,e1 (0, lnK)

−Ge2,e1−e2,e2 (0, lnK)+KG0,e1−e2,e2 (0, lnK) .

Î “Put on min,” giving the holder the right to sell the minimum of two assets ata given strike at expiry. The payoff here is

max(K−min(S1T ,S2T ),0)= (K−min(S1T ,S2T ))1[K>min(S1T ,S2T )]

= (K−S1T )1[K>S1T ]1[S1T<S2T ]+(K−S2T )1[K>S2T ]1[S1T>S2T ].

The price of this option at time t is obtained by adding four quantities:

Et[exp(−r f (T − t)

)max(max(S1T ,S2T )−K,0)

]= −Ge1,e1−e2,e1 (0, lnK)+KG0,e1−e2,e1 (0, lnK)

−Ge2,e2−e1,e2 (0, lnK)+KG0,e2−e1,e2 (0, lnK) .

Ï “Exchange one asset for another and earn the spread between the two.” Thepayoff here is

max(S1T −S2T ,0)= (S1T −S2T )1[S1T>S2T ].

The price of this option at time t is the difference of two quantities:

Et[exp(−r f (T − t)

)max(S1T −S2T ,0)

]= Ge1,e2−e1,0 (0,0)−Ge2,e2−e1,0 (0,0) .

15

5 Data, estimation and options pricingIn this section, we describe the data used in our empirical exercise. We estimatemodel parameters by maximizing the likelihood function and use the estimates asinputs in pricing rainbow contracts.

5.1 Data

For the empirical illustration, we use return data from 8 March 1971 to 14 August2013 on the NASDAQ and NYSE composite indices, among the most importantfinancial indices. It is worth noting that the two indices reflect the performanceof two quite distinct markets. In particular, NASDAQ loads on technology stocks,such as Microsoft and Intel, whereas NYSE contains a large proportion of mostlywell-established large industrial companies, such as General Electric and Ford.

Figure 1 shows the time series of realized volatility for the two indices.The period under consideration covers the 1973 oil crisis, the stock market crashof October 1987, the dot-com bubble burst of the early 2000s and the most recentfinancial crisis. From a purely descriptive point of view, the realized volatility andthe realized correlation between the two indices clearly show that they share somewell-documented, stylized facts about volatility. In particular, the time series of thevolatility of these indices is found to be consistent with time-varying volatility aswell as the volatility clustering effect. In addition, both series share similar patternsduring the most recent financial crisis, which has unsurprisingly led to a sustainedperiod of higher volatilities and correlations. Nevertheless, the figure also revealsthat the NYSE and NASDAQ indices can also react rather differently to some kindsof extreme events. In fact, the NASDAQ market was hit more strongly by thebursting of the high-tech bubble in the early 2000s, whereas the crash of October1987 mostly affected the NYSE stocks.

5.2 Estimation and historical model analysis

Parameters are estimated through the maximum likelihood procedure and reportedin Table 1. All the parameters are significant (see the standard errors in parenthe-ses); to avoid identification issues, we impose matrices α , β and σ to be lowertriangular. The Wishart distribution degrees-of-freedom parameter n is set to 2 andthe variance matrix V to 1/p times the identity matrix (V = (1/p)In).

As expected, the variance-covariance matrix is persistent as eigenvalues ofββ ′+αα ′ are close to 1. The prices of risk (λ ) are all positive and have the ex-pected sign, and thus market participants expect positive returns for risks taken.

16

The NASDAQ’s λ is higher than that of the NYSE, which means that market par-ticipants expect more reward for risks taken in the technology stocks compared toindustrial companies. The δ measures the correlation between a shock in returnsand a shock in realized variances, and gauges the so-called leverage effect. A neg-ative δ indicates a negative instantaneous correlation between returns and realizedvariances. The δ of the NYSE is negative, as expected, while that of NASDAQ issurprisingly positive, which implies that periods of high volatility in the technologysector coincide with periods of high returns.

Table 1: Estimation of the MARV model on daily NASDAQ and NYSE returns andRV (1971-2013)

λ 1.361 1.077 δ 5.025 -0.907(5.19E−01) (6.17E−02) (1.87E +00) (1.49E +00)

ω 2.97E-05 2.05E-05 α 5.83E-07 0.00E+00(3.84E−07) (3.34E−07) (2.85E−08)

2.05E-05 2.68E-05 5.30E-07 -1.62E-06(3.34E−07) (2.79E−07) (3.48E−08) (2.63E−07)

β 0.971 0.000 σ 3.91E-03 0.00E+00(2.27E−04) (0.00E +00) (1.51E−06) (0.00E +00)

0.000 0.966 2.70E-03 2.56E-03(2.94E−04) (3.28E−04) (1.63E−06) (7.74E−07)

The key ingredient in multivariate options pricing is the conditional variance-covariance matrix. How well our model is able to price options accurately dependson how accurately our model forecasts the future realized covariance matrix. Giventhat the conditional variance-covariance matrix is also the conditional expectationof future realized covariance, the model fit can be evaluated by comparing the ex-ante measure with the ex-post. Figure 2 compares the time series of both the condi-tional volatilities and correlations extracted from our affine model with the observedrealized volatilities and correlations. Our model’s conditional moments forecast ac-curately the ex-post moments.

We exploit the well-established tractability and flexibility properties of theaffine processes, raising the natural question of whether the cost to get the closed-form price (and, thus, to have an affine model) is too high. To answer this question,

17

we consider the dynamic of the non-affine conditional correlation model (DCC)of Engle (2000), which is the benchmark for conditional covariance matrix mod-elling. Options pricing in the DCC can only be done through simulations. Figure3 compares the time series of both the conditional volatilities and correlations ex-tracted from our affine model with those of the DCC ones. Despite its affine nature,our model’s conditional moments are very close to those of the DCC and we expectthe two models to generate similar options prices.

In the following section, we provide empirical illustrations that demonstratethe real potential of our approach in options pricing.

5.3 Options pricing

In this subsection, we compute options prices using two methodologies, namelyMonte Carlo simulation and our closed-form formula. Note, though, that the goalhere is to show how our pricing formula really works in practice. We consider rain-bow options with three different maturities: 1, 2 and 3 months. For each of thesematurities we consider five different strike prices, from 80 to 120, in incrementsof 10. Moreover, to put the two indices on an equal footing, it is assumed that thestarting values are 100 for both indices. This has the effect of essentially consider-ing options on the worst-performing of the indices. We compute the prices on eachof the last 100 days in our sample.

18

Table 2: Average (across time) options prices

Moneyness (S/X)Maturity (in months) 1.25 1.11 1.00 0.91 0.83

Bests of assets1M 103.33 103.60 105.70 111.53 120.262M 105.50 106.34 109.01 114.23 121.693M 107.44 108.69 111.60 116.54 123.41

Call on max1M 23.33 13.60 5.70 1.53 0.262M 25.50 16.34 9.01 4.23 1.693M 27.44 18.69 11.60 6.54 3.41

Call on min1M 17.59 8.50 2.59 0.46 0.052M 16.92 9.12 4.01 1.44 0.433M 16.94 9.88 5.10 2.35 0.99

Put on max1M 0.01 0.28 2.38 8.21 16.952M 0.14 0.99 3.65 8.86 16.333M 0.39 1.64 4.56 9.52 16.36

Put on min1M 0.06 0.97 5.07 12.93 22.522M 0.63 2.84 7.73 15.15 24.163M 1.45 4.39 9.60 16.85 25.49

Table 2 reports the average (across the time-series dimension) prices. As ex-pected, those prices generally increase with maturity. In the money call (S/X > 1),contracts are much more expensive than out of the money, while out of the moneyputs (S/X > 1) are cheaper than in the money. "Put on min" contracts are gener-ally less expensive than those for "Put on max," and the same result applies to callcontracts. This is as expected, since the premium in the "Put on min" is lower thanthat of the "Put on max." Beyond simply pricing options, we make clear throughoutthe following subsection that our results provide valuable loading insights by whichoptions market participants bet on specific factors of the returns dynamic.

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5.4 Disentangling our closed-form option formula

To gain additional insights into our pricing formula, we next decompose the overallprice into three components: the "Constant-Part" that does not require any integral(see equation (16)), the "Volatility-Part" that requires computing a univariate inte-gral (see equation (17)) and finally the "Correlation-Part" that requires computinga double integral (see equation (18)). We study the main drivers in the time-seriesmotion of all three components. To our surprise, there is almost no time-series vari-ation in equation (16), which implies that changes in the variance-covariance matrixhave no effect on that equation. For that reason we designate it the "Constant-Part,"since it primarily drives only the level of the contract price.

Table 3: Correlation (across time) between the “Volatility-Part” of options pricesand the average conditional volatility

Moneyness (S/X)Maturity (in months) 1.25 1.11 1.00 0.91 0.83

Bests of assets1M 0.95 0.99 0.99 0.99 0.992M 0.95 0.99 0.99 0.99 0.993M 0.96 0.99 0.99 0.99 0.99

Call on max1M 0.95 0.99 0.99 0.99 0.992M 0.95 0.99 0.99 0.99 0.993M 0.96 0.99 0.99 0.99 0.99

Call on min1M 0.98 1.00 1.00 1.00 1.002M 0.98 1.00 1.00 1.00 1.003M 0.98 1.00 1.00 1.00 1.00

Put on max1M 0.98 1.00 1.00 1.00 1.002M 0.98 1.00 1.00 1.00 1.003M 0.98 1.00 1.00 1.00 1.00

Put on min1M 0.95 0.99 0.99 0.99 0.992M 0.95 0.99 0.99 0.99 0.993M 0.96 0.99 0.99 0.99 0.99

20

Table 3 reports the correlation (across time) between the "Volatility-Part"of options prices and the average volatility level. Looking across maturities, mon-eyness and contract types, we see that the correlation between the average (acrossassets) of volatilities and the volatility component of options prices is almost 1.That part of the contract price is the cost paid by options buyers to hedge againsttime variations in individual stock volatilities.

Table 4: Correlation (across time) between the “Correlation-Part” of options pricesand the conditional correlation

Moneyness (S/X)Maturity (in months) 1.25 1.11 1.00 0.91 0.83

Bests of assets1M -0.97 0.80 0.87 0.88 0.852M -0.98 0.75 0.86 0.87 0.873M -0.98 0.68 0.85 0.87 0.88

Call on max1M -0.87 -0.89 -0.83 0.90 0.892M -0.95 -0.97 -0.96 0.97 0.973M -0.97 -0.98 -0.98 0.98 0.98

Call on min1M 0.87 0.89 0.83 -0.90 -0.892M 0.95 0.97 0.96 -0.97 -0.973M 0.97 0.98 0.98 -0.98 -0.98

Put on max1M -0.87 -0.89 -0.83 0.90 0.892M -0.95 -0.97 -0.96 0.97 0.973M -0.97 -0.98 -0.98 0.98 0.98

Put on min1M 0.87 0.89 0.83 -0.90 -0.892M 0.95 0.97 0.96 -0.97 -0.973M 0.97 0.98 0.98 -0.98 -0.98

Table 4 reports the correlation (across time) between the "Correlation-Part"of options prices and the conditional correlation. Looking across maturities, mon-eyness and contract types, we see that the co-movement between the conditionalcorrelation and the "Correlation-Part" of options prices is very high. That part of

21

the contract price is the cost paid by options buyers to hedge against time variationsin correlations among stocks.

Overall, results suggest that our decomposition can shed light on some fun-damental drivers in options valuation.

5.5 Closed-form vs. simulated options prices

In this section, we carry out numerical experiments to illustrate the accuracy ofour multivariate closed-form options price by comparing it to the simulated optionsprice.13 Given computational budget constraints often encountered in business en-vironments, the model’s compliance with reasonable pricing time becomes an im-portant issue.

It should be stressed that as we increase the number of simulations, weclearly increase the computation time. To illustrate this point, we report in Table 5the computation times in function of the number of simulations, and sum the timerequired to compute all the contract prices, across all the dimensions. For 10,000paths, it takes approximately one hour. The running time for our pricing formula isabout five minutes.

Table 5: Options prices computation time (in min) by number of simulated paths(in thousands)

NSP Time1 5.952 13.243 23.644 30.185 37.816 43.647 53.088 58.389 60.3810 64.50Closed form 5.18

13The software used is Matlab 2013 on a 3.5 GHz computer with 16 GB of memory

22

Table 6: Average (across time) of the differences between simulated and closed-form options prices

Moneyness (S/X)Maturity (in months) 1.25 1.11 1.00 0.91 0.83

Bests of assets1M 0.05 0.05 0.03 -0.01 -0.012M -0.01 0.01 -0.01 -0.02 0.003M 0.10 0.05 0.02 0.02 0.01

Call on max1M 0.05 0.05 0.03 -0.01 -0.012M -0.01 0.01 -0.01 -0.02 0.003M 0.10 0.05 0.02 0.02 0.01

Call on min1M -0.01 0.01 0.01 -0.01 -0.002M -0.06 -0.04 -0.01 -0.02 -0.013M 0.04 0.02 0.02 0.03 0.00

Put on max1M 0.00 -0.00 -0.01 -0.02 -0.062M 0.00 -0.01 -0.01 -0.01 0.013M -0.01 -0.01 0.00 -0.01 0.01

Put on min1M 0.00 -0.00 -0.00 -0.01 0.012M -0.00 0.01 -0.01 0.00 -0.013M 0.00 -0.01 -0.01 -0.01 0.02

Table 6 reports the difference between the simulated and our proposed op-tions prices, both computed from the ARV model. The main point from this tableis that both options prices are similar, which reinforces the validity of our theory.

6 ConclusionThis paper extends the single-condition derivatives pricing framework of Duffieet al. (2000), based on the Fourier transform. Options pricing formulas are givenup to numerical integrations. Our approach allows for quite general aggregatecontingent claims pricing with several sources of randomness (including stochas-

23

tic volatility and jump). Our theoretical methodology provides a valuable tool inthe options pricing literature. It is noteworthy that a significant improvement overMonte Carlo in computational efficiency is attained without sacrificing pricing ac-curacy. Moreover, this paper disentangles options prices into intuitive componentsthat enable traders to adequately assess their exposures to each options price driver.

The paper highlights some interesting methodological aspects of the Fourierinversion formula in a highly stylized options pricing setting, but it leaves out sev-eral real-world issues such as risk-neutral parameters calibration and risk-neutraldistribution fitting. Our approach would be useful in these and other areas as moreoptions data become available.

24

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Navatte, P. and C. Villa (2000): “The information content of implied volatility,skewness and kurtosis: empirical evidence from long-term cac 40 options,” Eu-ropean Financial Management, 6, 41–56, URL http://dx.doi.org/10.1111/1468-036X.00110.

Qu, D. (2010): “Pricing basket options with skew,” Wilmott Magazine.Rota, G.-C. (1964): “On the foundations of combinatory theory of Möbius func-

tions,” Wahrscheinlichkeitstheorie und Verw, 2, 340–368.Shephard, N. and K. Sheppard (2010): “Realising the future: forecasting with high-

frequency-based volatility (heavy) models,” Journal of Applied Econometrics,25, 197–231, URL http://dx.doi.org/10.1002/jae.1158.

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van Binsbergen, J., M. Brandt, and R. Koijen (2012): “On the timing and pricing ofdividends,” American Economic Review, 102, 1596–1618, URL http://www.aeaweb.org/articles.php?doi=10.1257/aer.102.4.1596.

27

A Proof of Proposition 2For 0 < τ1,τ2 <+∞ and y1,y2 ∈ R, let us define the expression I by the followingrelation:

I≡∫

τ1−τ1

∫τ2−τ2

eiv2y2

iv2

[eiv1y1ψ(a− iv1b1− iv2b2,x, t,T )− e−iv1y1ψ(a+ iv1b1− iv2b2,x, t,T )

iv1

]−e−iv2y2

iv2

[eiv1y1ψ(a− iv1b1 + iv2b2,x, t,T )− e−iv1y1ψ(a+ iv1b1 + iv2b2,x, t,T )

iv1

]dv1dv2.

(19)Because ψ(a+ iv1b1 + iv2b2,x, t,T ) =

∫R2 eiv1z1+iv2z2Ga,b1,b2(dz1,dz2;x,T ), the re-

lation (19) can be expressed as

I=∫

τ1

−τ1

∫τ2

−τ2

∫R2

eiv1y1+iv2y2−iv1z1−iv2z2− e−iv1y1+iv2y2+iv1z1−iv2z2

(iv1)(iv2)

−eiv1y1−iv2y2−iv1z1+iv2z2 + e−iv1y1−iv2y2+iv1z1+iv2z2

(iv1)(iv2)Ga,b1,b2(dz1,dz2;x,T )dv1dv2.

Since we disregard the exact order in which the integral has been composed, inte-grand permutations conserve the value of the integral, and I can be expressed asfollows:

I=∫R2

∫τ1

−τ1

∫τ2

−τ2

eiv1y1+iv2y2−iv1z1−iv2z2− e−iv1y1+iv2y2+iv1z1−iv2z2

(iv1)(iv2)

−eiv1y1−iv2y2−iv1z1+iv2z2 + e−iv1y1−iv2y2+iv1z1+iv2z2

(iv1)(iv2)dv1dv2Ga,b1,b2(dz1,dz2;x,T ).

(20)

To proceed with the determination of I, we make use of the following relation. Forτ1 > 0,τ2 > 0,

∫τ1−τ1

∫τ2−τ2

eiv1y1+iv2y2−iv1z1−iv2z2− e−iv1y1+iv2y2+iv1z1−iv2z2− eiv1y1−iv2y2−iv1z1+iv2z2 + e−iv1y1−iv2y2+iv1z1+iv2z2

(iv1)(iv2)dv1dv2

=∫

τ1−τ1

∫τ2−τ2

eiv1y1+iv2y2−iv1z1−iv2z2 + e−iv1y1−iv2y2+iv1z1+iv2z2

iv1iv2− e−iv1y1+iv2y2+iv1z1−iv2z2 + eiv1y1−iv2y2−iv1z1+iv2z2

iv1iv2dv1dv2.

(21)

28

Using the usual trigonometric identity, (21) becomes (22), as follows:∫τ1−τ1

∫τ2−τ2

eiv1y1+iv2y2−iv1z1−iv2z2− e−iv1y1+iv2y2+iv1z1−iv2z2− eiv1y1−iv2y2−iv1z1+iv2z2 + e−iv1y1−iv2y2+iv1z1+iv2z2

(iv1)(iv2)dv1dv2

=∫

τ1−τ1

∫τ2−τ2−2cos(v1(y1− z1)+ v2(y2− z2))−2cos(v1(y1− z1)− v2(y2− z2))

v1v2dv1dv2

=∫

τ1−τ1

∫τ2−τ2

2[

cos(v1(y1− z1)− v2(y2− z2))− cos(v1(y1− z1)+ v2(y2− z2))

v1v2

]dv1dv2

=∫

τ1−τ1

∫τ2−τ2

2[

cos(v1(y1− z1))cos(v2(y2− z2))+ sin(v1(y1− z1))sin(v2(y2− z2))

v1v2

]−2[

cos(v1(y1− z1))cos(v2(y2− z2))− sin(v1(y1− z1))sin(v2(y2− z2))

v1v2

]dv1dv2

=∫

τ1−τ1

∫τ2−τ2

4sin(v1(y1− z1))sin(v2(y2− z2))

v1v2dv1dv2

= 4∫

τ1−τ1

sin(v1(y1− z1))

v1dv1∫

τ2−τ2

sin(v2(y2− z2))

v2dv2.

(22)Furthermore, the following result holds for every τ > 0:∫

τ

−τ

e−iv(z−y)− eiv(z−y)

ivdv =

∫τ

−τ

−2sin(v(z− y))v

dv =−2sgn(z− y)∫

τ

−τ

−sin(v|z− y|)v

dv.

(23)

And limτ→∞

∫τ

τ

sin(v|z− y|)v

dv = π.

Pooling together the relation (23) with the bounded convergence theorem and usingthe fact that lim(y1,y2)→(∞,∞)Ga,b1,b2(y1,y2;x,T ) = ψ(a,x,0,T ), I as defined in (19),when letting τ1,τ2,→ ∞, this brings about

lim(τ1,τ2)→(∞,∞)) I4π2 =

∫R2 sgn(z1− y1)sgn(z2− y2)Ga,b1,b2(dz1,dz2;x,T )

=∫ +∞

y1

∫ +∞

y2

Ga,b1,b2(dz1;dz2)︸ ︷︷ ︸I

−∫ +∞

y1

∫ y2

−∞

Ga,b1,b2(dz1;dz2)︸ ︷︷ ︸II

−∫ +∞

y2

∫ y1

−∞

Ga,b1,b2(dz1;dz2)︸ ︷︷ ︸III

+

∫ y1

−∞

∫ y2

−∞

Ga,b1,b2(dz1;dz2)︸ ︷︷ ︸IV

.

(24)We determine each of the expressions above in order to compute I. Firstly, weproceed by computing the expression I

I=∫ +∞

y1

∫ +∞

y2Ga,b1,b2(dz1;dz2) =

∫∞

y1

[Ga,b1,b2(dz1;+∞)−Ga,b1,b2(dz1;y2)

]=

∫∞

y1Ga,b1,b2(dz1;+∞)−

∫∞

y1Ga,b1,b2(dz1;y2)

= Ga,b1,b2(+∞;+∞)−Ga,b1,b2(y1;+∞)−Ga,b1,b2(+∞;y2)+Ga,b1,b2(y1;y2).

We know that, whenever one of y1 or y2 is +∞, we recover Duffie et al. (2000)one-condition framework; then, from their Proposition 2, we have

Ga,b1,b2(y1;+∞) =ψ(a,x, t,T )

2+

14π

∫ +∞

−∞

eivy1ψχ(a− ivb1,x, t,T )− e−ivy1ψ(a+ ivb1,x, t,T )iv

dv

Ga,b1,b2(+∞;y2) =ψ(a,x, t,T )

2+

14π

∫ +∞

−∞

eivy2ψχ(a− ivb2,x, t,T )− e−ivy2ψ(a+ ivb2,x, t,T )iv

dv.

29

Plugging these relations into I yields

I = Ga,b1,b2(y1;y2) − 14π

∫ +∞

−∞

eivy1ψ(a− ivb1,x, t,T )− eivy1ψ(a+ ivb1,x, t,T )iv

dv

− 14π

∫ +∞

−∞

eivy2ψ(a− ivb2,x, t,T )− eivy2ψ(a+ ivb2,x, t,T )iv

dv.(25)

The second expression, II, is derived as follows:

II=∫ +∞

y1

∫ y2−∞

∂ 2Ga,b1,b2(z1;z2) =∫ +∞

y1dGa,b1,b2(z1;y2−)

= Ga,b1,b2(+∞;y2−)−Ga,b1,b2(y1;y2−),

where Ga,b1,b2(−y1,y2,x,T ) = limz1→y1,z1<y1;z2→y2,z2>y2 Ga,b1,b2(z1,z2,x,T ) andGa,b1,b2(−y1,−y2,x,T ) = limz1→y1,z1<y1;z2→y2,z2<y2 Ga,b1,b2(z1,z2,x,T ).

From Proposition 2 of Duffie et al. (2000) we also have

Ga,b1,b2(+∞;y2−)=ψ(a,x, t,T )

2+

14π

∫ +∞

−∞

eivy2ψ(a− ivb2,x, t,T )− e−ivy2ψ(a+ ivb2,x, t,T )iv

dvandGa,b1,b2(y1;y2−) = Ga,b1,b2(y1;y2). The expression II then becomes

II =ψ(a,x, t,T )

2+

14π

∫ +∞

−∞

eivy2ψ(a− ivb2,x, t,T )− e−ivy2ψ(a+ ivb2,x, t,T )iv

dv−Ga,b1,b2(y1;y2).

(26)Likewise, III is computed as

III=∫ +∞

y2

∫ y1−∞

Ga,b1,b2(dz1;dz2) =∫ +∞

y2dGa,b1,b2(y1−;z2)

= Ga,b1,b2(y1−;+∞)−Ga,b1,b2(y1−;y2)

=ψ(a,x, t,T )

2+

14π

∫ +∞

−∞

eivy1ψ(a− ivb1,x, t,T )− e−ivy1ψ(a+ ivb1,x, t,T )iv

dv−Ga,b1,b2(y1;y2).

(27)Finally, the last term, IV , is expressed as

IV =

∫ y1

−∞

∫ y2

−∞

Ga,b1,b2(dz1;dz2) =

∫ y1

−∞

dGa,b1,b2(z1;y2−) = Ga,b1,b2(y1−;y2−) = Ga,b1,b2(y1;y2).

(28)In conclusion, we can express the following limit:

14π2 lim(τ1,τ2)→(∞,∞)) I = 4Ga,b1,b2(y1;y2)−ψ(a,x, t,T )− 1

2π

∫ +∞

−∞

eivy2ψ(a− ivb2,x, t,T )− eivy2ψ(a+ ivb2,x, t,T )iv

dv

− 12π

∫ +∞

−∞

eivy1ψ(a− ivb1,x, t,T )− e−ivy1ψ(a+ ivb1,x, t,T )iv

dv.

(29)

B Proof of Lemma 1The proof of Lemma 1 relies on the following combinatory identity:

30

∏j∈A

(r j + s j

)=∑B⊆A

∏j∈B

r j

∏j∈A\B

s j

. (30)

Substituting r j→−eiv j(y j−z j)

2i,s j→

e−iv j(y j−z j)

2iand using the fact that

sin(v j(y j− z j)

)=

eiv j(y j−z j)− e−iv j(y j−z j)

2i, we have

−2sin(v j(y j− z j)

)v j

=e−iv j(y j−z j)− eiv j(y j−z j)

iv jand (30) becomes

(−2)|A|∏j∈A

sin(v j(y j− z j)

)v j

=

∑B⊆A(−1)|B|eiv(B)(y(B)−z(B))−iv(A\B)(y(A\B)−z(A\B))∏

j∈A(iv j).

(31)We therefore show equation (11) of Lemma 1 by combinatory identity.

Substituting QA by its expression (11) in IA leads to

IA =∫R|A|∫R|A| (−2)|A|

|A|∏j=1

sin(v j(y j−z j))v j

∏j∈A dv jGa,b(dz(A),x,T )

= (−2)|A|∫R|A|∏

j∈A

[∫ sin(v j(y j−z j))v j

dv j

]Ga,b(dz(A),x,T )

= (−2π)|A|∫R|A|∏

j∈A sgn(z j− y j

)Ga,b(dz(A),x,T ).

C Proof of Lemma 2Henceforth, to simplify our notations, we drop the multiple occurrences of Xt ,T,χfrom the expression of Ga,b(z,Xt ,T ). By definition, we have

IA =

∫R|A|

∑|A|k=0(−1)k∑

Ak⊆A eiv(Ak)y(Ak)−iv(A\Ak)

y(A\Ak)ψ(a− iv(Ak)b(Ak)+ iv(A\Ak)b(A\Ak)

)∏j∈A(iv j)

∏j∈A

dv j.

Using the fact that

ψ(a− iv(Ak)b(Ak)+ iv(A\Ak)b(A\Ak)

)=

∫R|A|

e−iv(Ak)z(Ak)

+iv(A\Ak)z(A\Ak)Ga,b(dz(A),Xt ,T ),

31

we have

IA

=∫R|A|

∑|A|k=0(−1)k∑

Ak⊆A eiv(Ak)y(Ak)−iv(A\Ak)

y(A\Ak)∫R|A| e

−iv(Ak)z(Ak)

+iv(A\Ak)z(A\Ak)Ga,b(dz(A))∏

j∈A(iv j)

∏j∈A dv j

=∫R|A|∫R|A|

∑|A|k=0(−1)k∑

Ak⊆A eiv(Ak)(y(Ak)−z(Ak))−iv(A\Ak)(y(A\Ak)

−z(A\Ak))∏j∈A(iv j)

∏j∈A dv jGa,b(dz(A))

=∫R|A|∫R|A| QA

∏j∈A dv jGa,b(dz(A),Xt ,T ),

where

QA ≡∑|A|

k=0(−1)k∑Ak⊆A eiv(Ak)

(y(Ak)−z(Ak)

)−iv(A\Ak)

(y(A\Ak)

−z(A\Ak)

)∏

j∈A(iv j).

Replacing QA by its expression in IA leads to

IA =∫R|A|∫R|A| (−2)|A|

|A|∏j=1

sin(v j(y j−z j))v j

∏j∈A dv jGa,b(dz(A),Xt ,T )

= (−2)|A|∫R|A|∏

j∈A

[∫ sin(v j(y j−z j))v j

dv j

]Ga,b(dz(A),Xt ,T )

= (−2π)|A|∫R|A|∏

j∈A sgn(z j− y j

)Ga,b(dz(A),Xt ,T ).

We are now ready to demonstrate the lemma. The goal is to show that

IA = (−2π)|A||A|∑

k=0

(−2)k∑Ak⊆A

Ga,b(y(Ak)).

In other words, we endeavor to prove that

JA ≡∫R|A|

∏j∈A

sgn(z j− y j

)Ga,b(dz(A)) =

|A|∑k=0

(−2)k∑Ak⊆A

Ga,b(y(Ak)). (32)

To do so, we proceed by induction. Suppose that the relation holds for each subsetof E whose member’s cardinality is less than or equal to |A|. Let us show that the

32

relation is also fulfilled for the subset with |A|+ 1 elements. Denoting A+ 1 ≡A∪{y|A|+1}, we have

JA+1 =∫R|A|+1

∏j∈A+1 sgn

(z j− y j

)Ga,b(dz(A+1))

=∫R sgn(z|A|+1− y|A|+1)d

[∫R|A|∏

j∈A sgn(z j− y j

)Ga,b(dz(A);z|A|+1)

].

By induction, we can apply (32) to∫R|A|∏

j∈A sgn(z j− y j

)Ga,b(dz(A);z|A|+1). This

implies that

∫R|A|

∏j∈A

sgn(z j− y j

)Ga,b(dz(A);z|A|+1) =

|A|∑k=0

(−2)k∑Ak⊆A

Ga,b(y(Ak);z|A|+1), (33)

thus

JA+1 =

∫R

sgn(z|A|+1− y|A|+1)

|A|∑k=0

(−2)k∑Ak⊆A

Ga,b(y(Ak);dz|A|+1)

,hence

JA+1

=∑|A|

k=0(−2)k∑Ak⊆A

∫R sgn(z|A|+1− y|A|+1)Ga,b

(y(Ak);dz|A|+1

)=

∑|A|k=0(−2)k∑

Ak⊆A

[−∫ y|A|+1−∞ Ga,b

(y(Ak);dz|A|+1

)+∫

∞

y|A|+1Ga,b

(y(Ak);dz|A|+1

)]=

∑|A|k=0(−2)k∑

Ak⊆A

[−2Ga,b

(y(Ak);y|A|+1

)+Ga,b

(y(Ak);−∞

)+Ga,b

(y(Ak);∞

)].

By the definition of Ga,b(·), we have

Ga,b(y(Ak);−∞

)= 0, Ga,b

(y(Ak);∞

)= Ga,b

(y(Ak)

),

and therefore

JA+1 =

|A|∑k=0

(−2)k∑Ak⊆A

[−2Ga,b

(y(Ak);y|A|+1

)+Ga,b

(y(Ak)

)]=

|A|∑k=0

(−2)k+1∑Ak⊆A

Ga,b(y(Ak);y|A|+1

)+JA

= (−2)|A|+1Ga,b(y(A+1))+

|A|−1∑k=0

(−2)k+1∑Ak⊆A

Ga,b(y(Ak);y|A|+1

)+JA.

33

Replacing∑

Ak⊆A Ga,b(y(Ak);y|A|+1

)by

∑Ak⊆A

Ga,b(y(Ak);y|A|+1

)=

[ ∑Ak⊆A Ga,b

(y(Ak);y|A|+1

)+∑

Ak+1⊆A Ga,b(y(Ak+1)

)−∑

Ak+1⊆A Ga,b(y(Ak+1)

) ]implies that

JA+1 =(−2)|A|+1Ga,b(y(A+1))+

|A|−1∑k=0

(−2)k+1[ ∑

Ak⊆A Ga,b(y(Ak);y|A|+1

)+∑

Ak+1⊆A Ga,b(y(Ak+1)

)−∑

Ak+1⊆A Ga,b(y(Ak+1)

) ]+JA.

Hence

JA+1 = (−2)|A|+1Ga,b(y(A+1))+

|A|−1∑k=0

(−2)k+1

∑Ak⊆A

Ga,b(y(Ak);y|A|+1

)+∑

Ak⊆A+1

Ga,b(y(Ak)

)−|A|−1∑k=0

(−2)k+1

∑Ak+1⊆A

Ga,b(y(Ak+1)

)+JA.

By noting that∑Ak⊆A

Ga,b(y(Ak);y|A|+1

)+∑

Ak⊆A+1

Ga,b(y(Ak)

)=

∑A j⊆A+1

Ga,b

(y(A j)

),

we can rewrite JA+1 as the following:

JA+1 = (−2)|A|+1Ga,b(y(A+1))+

|A|−1∑k=0

(−2)k+1∑

Ak+1⊆A+1

Ga,b(y(Ak+1)

)

−|A|−1∑k=0

(−2)k+1

∑Ak+1⊆A

Ga,b(y(Ak+1)

)+JA.

Rearranging the summation leads to

JA+1 = (−2)|A|+1Ga,b(y(A+1))+

|A|∑j=1

(−2) j∑

A j⊆A+1

Ga,b

(y(A j)

)

−|A|−1∑k=0

(−2)k+1

∑Ak+1⊆A

Ga,b(y(Ak+1)

)+JA,

which implies

JA+1 =

|A|+1∑j=1

(−2) j∑

A j⊆A+1

Ga,b

(y(A j)

)+JA−

|A|∑j=1

(−2) j

∑A j⊆A

Ga,b

(y(A j)

) .

34

Using the induction assumption up to JA, we have

JA−|A|∑j=1

(−2) j

∑A j⊆A

Ga,b

(y(A j)

)= Ga,b(y( /0))= ψ(a,x, t,T ).

Thus

JA+1 =

|A|+1∑j=1

(−2) j∑

A j⊆A+1

Ga,b

(y(A j)

)+ψ(a,x, t,T ).

In other words,

JA+1 =

|A|+1∑j=0

(−2) j∑

A j⊆A+1

Ga,b

(y(A j)

).

This ends the proof of Lemma 2.

D Proof of Proposition 3The proof of Proposition 3 relies on Lemma 2 and an application of the Möbius inversionfor the Boolean algebra of subsets of a finite set (see Hazewinkel, 2002; Rota, 1964).

Define A⊆ E,

G(A) = 2|A|Ga,b(y(A)), F(A) = (−2π)−|A|IA, (34)

where IA,A⊆ E is given by (10).Lemma 2 implies that

F(A) =∑B⊆A

(−1)|A|−|B|G(B). (35)

The Möbius inversion says that (35) is equivalent to

G(A) =∑B⊆A

F(B) ∀B⊆ A, (36)

and the proof of Proposition 3 is completed with A = E in (36).

E Some ARV model featuresBelow we provide some ARV model properties.

35

E.1 The density

The joint conditional density of returns (Rt) and realized variance-covariance matrix (RVt)is

ft−1 (Rt ,RVt) = ft−1 (Rt |RVt) ft−1 (RVt)

ft−1 (Rt |RVt) = (2π)−n2 |RVt |−

12 exp

(−1

2z′1,tz1,t

),

wherez1,t = RV−1/2

t [Rt −µt−1− (RVt −Σt−1)δ ] ,

and

ft−1 (RVt) = 2−pn/2 ∣∣σV σ′∣∣−p/2

Γn

( p2

)−1|Wt |

p−n−12 exp

−Tr((σV σ ′)−1Wt

)2

,

whereWt = σ

−1 (RVt −Σt−1)(σ−1)′+ pV.

E.2 Model characteristics

It follows from (14) that the conditional variance can be expressed as

Σt = ω +βΣt−1β′+αRVtα

′

= ω +βΣt−1β′+α

{Σt−1 +σ [Wt − pV ]σ ′

}α′

= ω +βΣt−1β′+αΣt−1α

′+ασWtσ′α′− pασV σ

′α′

= ω− pασV σ′α′+βΣt−1β

′+αΣt−1α′+ασWtσ

′α′.

Given thatΣt = ω− pασV σ

′α′+βΣt−1β

′+αΣt−1α′+ασWtσ

′α′,

the model is well defined (in the sense that the support of distribution of Σt is the symmetricpositive definite real matrix) whenever the following condition is fulfilled:

ω− pασV σ′α′ ≥ 0,

and we can express ω asω = pασV σ

′α′+ γγ

′.

Further, given thatEt−1 [Σt ] = ω +βΣt−1β

′+αΣt−1α′,

the variance matrix is covariance-stationary if all the eigenvalues of ββ ′+αα ′ are less than1.

36

E.3 Proof of Proposition 4, the conditional expectation of logreturns

Et−1[exp(e′iRt

)] = exp

(r f +λie′iΣt−1ei

)Et−1[exp

(e′iRt

)]

= Et−1

[exp(

e′iµt−1 + e′i (RVt −Σt−1)δ + e′iRV 1/2t z1,t

)]= Et−1

[exp(

e′iµt−1 + e′iσ [Wt − pV ]σ ′δ + e′iRV 1/2t z1,t

)]= Et−1

[exp(

e′iµt−1 + e′iσ [Wt − pV ]σ ′δ +12

e′iRVtei

)]= Et−1

[exp(

e′iµt−1 + e′iσ [Wt − pV ]σ ′δ

+12 e′i (Σt−1 +σ [Wt − pV ]σ ′)ei

)]= Et−1

[exp(

e′iµt−1 + e′iσ [Wt − pV ]σ ′δ

+12 (e′iΣt−1ei + e′iσ [Wt − pV ]σ ′ei)

)]

= Et−1

[exp(

e′iµt−1 + e′iσWtσ′δ − pe′iσV σ ′δ

+12 e′iΣt−1ei +

12 e′iσWtσ

′ei− p2 e′iσV σ ′ei

)]= Et−1

[exp(

e′iµt−1 +12 e′iΣt−1ei− pe′iσV σ ′δ − p

2 e′iσV σ ′ei

+e′iσWtσ′δ + 1

2 e′iσWtσ′ei

)]= Et−1

[exp(

e′iµt−1 +12 e′iΣt−1ei− pe′iσV σ ′δ − p

2 e′iσV σ ′ei

+Tr((

σ ′δe′iσ + 12 σ ′eie′iσ

)Wt) )]

= Et−1

[exp(

e′iµt−1 +12 e′iΣt−1ei− pe′iσV σ ′δ − p

2 e′iσV σ ′ei

− p2 ln[det(In−2

(σ ′δe′iσ + 1

2 σ ′eie′iσ)

V)] )]

.

Hence

e′iµt−1 = r f +

(λi−

12

)e′iΣt−1ei

+pe′iσV σ′δ +

p2

e′iσV σ′ei

+p2

ln[

det(

In−2(

σ′δe′iσ +

12

σ′eie′iσ

)V)]

.

E.4 Proof of the moment-generating functions

E.4.1 Proof of Proposition 5

Et−1[exp(u′Rt +Tr (θΣt)

)]= exp(A(u,θ)+Tr (B(u,θ)Σt−1))

37

with

A(u,θ) =

∑ni=1 ui

(r f + pe′iσV σ ′δ + p

2 e′iσV σ ′ei

+ p2 ln[det(In−2

(σ ′δe′iσ + 1

2 σ ′eie′iσ)

V)] )

− p2 ln[det(In−2

(12 σ ′uu′σ +σ ′δu′σ +σ ′α ′θασ

)V)]

−pu′σV σ ′δ − p2 u′σV σ ′u+Tr (θ (ω− pασV σ ′α ′))

B(u,θ) = β′θβ +α

′θα +

12

uu′+n∑

i=1

eiui

(λi−

12

)e′i.

In the following, we provide more details on the proof of that result:

Et−1[exp(u′Rt +Tr (θΣt)

)]= Et−1

[exp

(u′(

µt−1 +(RVt −Σt−1)δ +RV 1/2t z1,t

)+Tr [θ (ω− pασV σ ′α ′+βΣt−1β ′+αΣt−1α ′+ασWtσ

′α ′)]

)]

= Et−1

[exp(

u′ (µt−1 +σWtσ′δ − pσV σ ′δ )+ 1

2 u′RVtu+Tr [θ (ω− pασV σ ′α ′+βΣt−1β ′+αΣt−1α ′+ασWtσ

′α ′)]

)]= Et−1

[exp(

u′ (µt−1 +σWtσ′δ − pσV σ ′δ )+ 1

2 u′Σt−1u+ 12 u′σWtσ

′u− p2 u′σV σ ′u

+Tr [θ (ω− pασV σ ′α ′+βΣt−1β ′+αΣt−1α ′+ασWtσ′α ′)]

)]

= Et−1

exp

u′µt−1− pu′σV σ ′δ − p2 u′σV σ ′u+Tr (θ (ω− pασV σ ′α ′))

+Tr((

β ′θβ +α ′θα + 12 uu′

)Σt−1

)+Tr

[( 12 σ ′uu′σ +σ ′δu′σ +σ ′α ′θασ

)Wt]

= exp

u′µt−1− pu′σV σ ′δ − p2 u′σV σ ′u+Tr (θ (ω− pασV σ ′α ′))

+Tr((

β ′θβ +α ′θα + 12 uu′

)Σt−1

)− p

2 ln[det(In−2

( 12 σ ′uu′σ +σ ′δu′σ +σ ′α ′θασ

)V)]

= exp

∑ni=1 uie′iµt−1− pu′σV σ ′δ − p

2 u′σV σ ′u+Tr (θ (ω− pασV σ ′α ′))

+Tr((

β ′θβ +α ′θα + 12 uu′

)Σt−1

)− p

2 ln[det(In−2

(12 σ ′uu′σ +σ ′δu′σ +σ ′α ′θασ

)V)]

= exp

∑n

i=1 ui

(r f +

(λi− 1

2

)e′iΣt−1ei + pe′iσV σ ′δ + p

2 e′iσV σ ′ei

+ p2 ln[det(In−2

(σ ′δe′iσ + 1

2 σ ′eie′iσ)

V)] )

−pu′σV σ ′δ − p2 u′σV σ ′u+Tr (θ (ω− pασV σ ′α ′))

+Tr((

β ′θβ +α ′θα + 12 uu′

)Σt−1

)− p

2 ln[det(In−2

(12 σ ′uu′σ +σ ′δu′σ +σ ′α ′θασ

)V)]

= exp

∑n

i=1 ui

(r f + pe′iσV σ ′δ + p

2 e′iσV σ ′ei

+ p2 ln[det(In−2

(σ ′δe′iσ + 1

2 σ ′eie′iσ)

V)] )

− p2 ln[det(In−2

( 12 σ ′uu′σ +σ ′δu′σ +σ ′α ′θασ

)V)]

−pu′σV σ ′δ − p2 u′σV σ ′u+Tr (θ (ω− pασV σ ′α ′))

+Tr[(

β ′θβ +α ′θα + 12 uu′+

∑ni=1 eiui

(λi− 1

2

)e′i)

Σt−1]

.

38

E.4.2 Proof of Proposition 6

Et

[exp

(u′

τ+1∑i=1

Rt+i

)]= Et

[exp(u′Rt+1

)Et+1

[exp

(u′

τ+1∑i=2

Rt+i

)]]

= Et

exp(u′Rt+1

)Et+1

exp

u′τ∑

j=1

Rt+1+i−1

= Et

[exp(u′Rt+1 +C (u;τ)+Tr (D(u;τ)Σt+1)

)]= exp(C (u;τ))Et

[exp(u′Rt+1 +Tr (D(u;τ)Σt+1)

)]= exp(C (u;τ)+A(u,D(u;τ))+Tr (B(u,D(u;τ))Σt)) .

39

1976 1982 1987 1992 1998 2003 2009 20130

20

40

60

80

100Daily NASDAQ and NYSE RV

NASDAQNYSE

1976 1982 1987 1992 1998 2003 2009 20130

0.2

0.4

0.6

0.8

1Realized Daily Correlation between NASDAQ and NYSE

Figure 1: Realized covariance. Daily measures of realized volatilities and correla-tion, from 8 March 1971 to 14 August 2013.

40

1976 1982 1987 1992 1998 2003 2009 20130

50

100Daily Realized and Conditional Vol: NASDAQ

RVExpected RV

1976 1982 1987 1992 1998 2003 2009 20130

50

100Daily Realized and Conditional Vol: NYSE

RVExpected RV

1976 1982 1987 1992 1998 2003 2009 20130

0.5

1Daily Realized and Conditional Correlation

Realized CorrelationExpected Correlation

Figure 2: Realized covariance vs. conditional covariance matrix. Daily mea-sures of realized and conditional volatilities, as well as correlation, from 8 March1971 to 14 August 2013.

41

1976 1982 1987 1992 1998 2003 2009 20130

50

100Daily Conditional Vol: NASDAQ

1976 1982 1987 1992 1998 2003 2009 20130

50

100Daily Conditional Vol: NYSE

1976 1982 1987 1992 1998 2003 2009 20130

0.5

1Daily Conditional Correlation

ARVDCC

ARVDCC

ARVDCC

Figure 3: Conditional covariance matrix: ARV vs. DCC. Daily measures ofconditional volatilities and correlation extracted from the ARV and DCC, from 8March 1971 to 14 August 2013.

42